Welcome back everyone. 
I'm Charles Clark, and these next few 
lectures we're going to looking at the 
problem solving the Schrödinger Equation. 
Now in the recent Broadway stage musical, 
"Curtains", a character says Putting on a 
musical must be the most fulfilling thing 
a person could ever do. 
Well, here at exploring quantum physics, 
we think that solving the Schrodinger 
equation is one of the most fulfilling 
things a person can ever do. 
and today you will be that person. 
We are going to start by. 
Actually solving the Schrodinger equation 
or guiding you to solve the Schrodinger 
equation this is tailored to people 
who've never solved that equation in 
their lives before. 
But we hope there will always be some 
information that's useful to people with 
much experience. 
Well let's see. 
Well here is the Schrodinger equation. 
You've seen it before. 
I h. 
Bar decided to use h psi. 
And let me emphasize again according to 
quantum mechanics, all the information 
that we are able to use to describe the 
system is contained in the wave function. 
And so, this The equation is first order 
in time because it, it, tells us how to 
determine the state at some later time 
from the state at a given time. 
So this equation should be thought of as 
an initial value problem in which we are 
given psi of t at some initial time t And 
then, we integrate this equation to 
find[UNKNOWN] of T plus[UNKNOWN]. 
Well, you might ask, what could possibly 
go wrong in such a simple scheme. 
And here is a little in line video to 
test your understanding. 
Well I guess you see from this in line 
video that 
The mechanics of the Schrodinger equation 
is apparently quite straightforward. 
If we have a wave function and a 
Hamiltonian operator we can just advance 
the equation gradually in time by 
repeated application of that operator. 
Well like much in life The difficulty 
really emerges with the details. 
So here's a recent paper, I, showing you 
my own work cause I do know something 
about it. 
an it's a calculation relevant to some 
atomic clock research involving the 
Ytterbium atom. 
you can go look at your periodic table. 
Ytterbium is at the end of the rare earth 
period has 70 electrons. 
Now let's just say, supposing we were 
doing a, a, you know, a calculation that 
took very explicit account of all the 
electrons in the turban atom, there are 
70 of those. 
And if we represented them on a, a 
spacial grid that was just ten points on 
a side, which is not a lot to represent 
the motion of a particle we get this 
impossibly large number to deal with. 
Now, it can be reduced by quite a bit. 
Due to symmetry, considerations. 
But it's still just totally infeasible. 
So, 
Much of the, the practical issues of 
solving the Schrodinger equation have to 
do with finding feasible schemes. 
Now let me emphasize that the full 
Schrodinger equation is explicitly time 
dependence. 
It's, the equation for solving the, for 
the evolution of quantum system time. 
And, there are many cases in which. 
You need to pursue it from that 
standpoint. 
Here, here's an example. 
It's actually a solution of a, of a 
non-linear Schrödinger equation that 
describes Bose-Einstein condensate, and 
it shows the breakup of a soliton in that 
condensate into a series of vortex rings. 
So to l-, to understand things like this 
You need to have a full, time-dependent 
description. 
On the other hand, as we have seen over 
an over again, there're a lot of 
interesting phenomena, such as the, you 
know, my favorite, the Balmer's spectrum 
of the hydrogen atom, that seem to be 
described by stationary quantum states. 
That is, these are states that have, a 
very simple time evolution. 
And they're persistent. 
They lead to, to sharp structures in, in 
spectra such as the transitions between 
levels. 
so for the rest of this, these lectures 
we're going to focus on the stationary 
state problem in which we take the time 
dependent Schrodinger equation and then 
look at the special case when the 
solution is an eigenfunction of the 
Hamiltonian operator. 
Okay, so just to remember the 
terminology, this is, this is an 
eigenvalue equation, or eigenfunction, 
and the The Hamiltonian operator is 
given, this is sort of like input to the 
calculation. 
Psi and E are unknown, Psi is a so called 
eigenfunction or eigenvector, we tend to 
use those terms exchangeably. 
And E, the energy of the system is the 
eigenvalue. 
Okay, so what does it mean to actually 
solve this equation? 
Well, the Schrodinger equation is 
implemented to describe a wide variety of 
physical systems. 
so our lectures are just going to focus 
on things that are described in terms of 
a, a potential that affects to motion of 
an electro-, a particle of mass m. 
So the, the general form that we're going 
to consider, which is very rich in terms 
of its applications Is an equation, it's 
a partial differential equation in three 
dimensions. 
we'll occasionally, we will of course for 
illustrative purposes consider the 
problems in lower dimensions as well. 
And so it has the form of the kinetic 
operator here minus h[UNKNOWN] squared 
over 2M times del squared. 
just so you remember, delta squared is d 
squared, dx squared plus d squared, dy 
squared plus d squared, dz squared. 
this is the Laplacian operator, widely 
used throughout physics, 
electromagnetism, acoustics. 
fluid dynamics, and quantum mechanics. 
V, is the potential soul[/g] for, for the 
hydrogen atom . 
V would be equal to minus Z E squared 
over R. 
Or for the harmonic oscillator 1/2 M mega 
squared R squared. 
So, for the usual type of problem, we 
know what the potential is, and we're 
kind of solved for the unknown wave 
function and energy. 
But we're going to start here with a, a 
constructive approach. 
Which is based on the idea. 
I mean, how many actual functions do you 
know? 
of course, you can say, okay. 
A function is X, or X plus X to the 5th 
over 7. 
You can generate a large number of ad hoc 
functions. 
but perhaps many of you have a fairly 
limited. 
Vocabulary in terms of generality that 
would be expanded during this course so 
lets just say we are going to start with 
functions that we recognize and then see 
if we can find a potential and an energy 
For which those functions are solutions 
of the Schrodinger equation. 
This is actually quite a straightforward 
thing to do. 
So here's the Schrodinger equation, now 
what we do is we take this term here and 
move it over to the right hand side. 
Then we take the Common factors of, of 
the wave function and put them over on, 
onto the far left here, and we get this 
very simple equation. 
Which means, I mean all you need to do to 
solve this, all you need to do is be able 
to take derivatives So you take the trial 
wave function, apply the Laplacian 
operator to it, divide by the wave 
function, and see what you get. 
So here's a little in-line quiz. 
To, help walk you through, to what must 
be the simplest possible example. 
So I hope you found that fairly 
straightforward. 
Now, the function that we c-, chose, 
which was, s-, size equal to a, a 
constant, is, you might say, it's not the 
most useful solution that you've ever 
seen. 
It does in fact have an application in 
scattering theory, but it's hardly a 
generic function. 
Our message for the moment is that any 
function will provide a solution to a 
Schrodinger equation. 
But not necessarily for physically 
interesting potential. 
And we'll see that in the next part that 
for any given value of the energy and 
whatever the potential may be, there are 
infinitely many solutions of this, this 
Schrodinger equation. 
So we need more information. 
Then is just, evident in the mathematical 
statement of the equation of this level 
in order to perform the solutions that we 
need to find. 
Having said that, however, we're going to 
conclude this part by construction of E 
and V for I say one of the most widely 
used functions in quantum mechanics, I 
think I'll argue in the next segment is 
the most widely used function in quantum 
mechanics, and you've seen it before but 
let's, let's look at it from the 
standpoint of the constructive approach. 
As act at this point switch to the online 
video, and online quiz and then we'll 
conclude with the a few-, the last few 
words I say, in a moment. 
So that was a demonstration of recovering 
the harmonic oscillator potential. 
From the Gaussian wave function, which is 
solution, and we're going to take that up 
in more detail in the next part.